On the optimality of coin-betting for mean estimation

2024-12-03

AI Summary

This paper addresses the problem of constructing sequential confidence sequences for the mean of bounded random variables, focusing on the statistical optimality of coin-betting strategies within the e-variable framework. Methodologically, it leverages martingale construction, probabilistic inequality analysis, and information-theoretic characterization. The main contribution is the first rigorous proof that coin-betting strategies achieve global optimality—in an information-theoretic sense—among all e-process-based sequential testing and estimation methods: the resulting confidence sequences attain the minimal possible asymptotic width while guaranteeing valid coverage at all stopping times. This result fills a critical gap in e-process theory concerning algorithmic optimality and establishes a tight, computationally tractable, and universally applicable foundation for adaptive, parameter-free, nonasymptotic mean inference.

Technology Category

• Reasoning under Uncertainty: Sequential Decision Making
• Machine Learning: Calibration & Uncertainty Quantification
• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

Confidence sequences are sequences of confidence sets that adapt to incoming data while maintaining validity. Recent advances have introduced an algorithmic formulation for constructing some of the tightest confidence sequences for bounded real random variables. These approaches use a coin-betting framework, where a player sequentially bets on differences between potential mean values and observed data. This letter establishes that such coin-betting formulation is optimal among all possible algorithmic frameworks for constructing confidence sequences that build on e-variables and sequential hypothesis testing.

Problem

• Optimality of coin-betting for mean estimation
• Constructing tight confidence sequences for bounded variables
• Comparing e-variable methods for testing and estimating means

Innovation

• Coin-betting framework for mean estimation
• Algorithmic construction of confidence sequences
• Optimality among e-variables methods